Analytic bridge diffusions for controlled path generation

Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrodinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad analytically solvable class in which, for a deterministic source and a Gaussian-mixture target, the score and all intermediate marginals are explicit and protocol objectives of the type used in this paper can be differentiated without inner stochastic simulation loops. We recast the classical linear--quadratic--Gaussian stochastic-control structure as a transport problem of the Path Integral Diffusion type. Linear dynamics, Gaussian noise, and quadratic running costs reduce the bridge calculation to a matrix Riccati cascade, while the terminal state cost is replaced by a prescribed Gaussian-Mixture terminal probability density. Linear Quadratic -- Gaussian Mixture -- Path Integral Diffusion (LQ-GM-PID) thereby turns bridge diffusion from terminal target matching alone into an analytically controlled laboratory for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance task, and a high-dimensional study reaching d=32 and M=16 terminal modes in separate scaling sweeps. We position LQ-GM-PID as an analytically solvable reference model in which score approximations, path-shaping objectives, and protocol-learning procedures can be tested against explicit quantities.